Calvert-Lewin’s 24th-minute goal proved decisive against a Hammers side who huffed and puffed but never quite hit their stride on a crucial afternoon, failing to register a single shot on target.
The closest West Ham came to the equalizer when Vladimir Coufal’s shot hit the post and Jarrod Bowen couldn’t cram into the rebound at close range.
Everton hit the post late thanks to Josh King and they hung on for the win which keeps them in the frame for a European place.
West Ham remained in fifth place with 58 points, five points behind Leicester City, fourth with three games to go. Liverpool are just one point away from West Ham with a match underway.
Everton are eighth with 55 points and have four games to go.